Derived invariance of the number of holomorphic $1$-forms and vector fields

Popa, Mihnea; Schnell, Christian

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 3, pp. 527-536.

Nous montrons que deux variétés projectives lisses dont les catégories dérivées sont équivalentes, ont des variétés de Picard isogènes. En particulier, elles ont la même irrégularité et le même nombre de champs de vecteurs indépendants. On en déduit l'invariance des nombres de Hodge par l'équivalence dérivée pour les variétés de dimension trois, ainsi que quelques autres conséquences numériques.

We prove that smooth projective varieties with equivalent derived categories have isogenous Picard varieties. In particular their irregularity and number of independent vector fields are the same. This implies that all Hodge numbers are the same for arbitrary derived equivalent threefolds, as well as other consequences of derived equivalence based on numerical criteria.

Classification : 14F05, 14K30
Keywords: derived categories, Picard variety, Hodge numbers
Mot clés : catégories dérivées, variété de Picard, nombres de Hodge

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